Dear Pedro, That’s an interesting point about “direct image” going back to sheaf theory. Bredon’s old book reminded me that the most natural definitions of f^* and f_* use quite different representations of sheaves. With local homeomorphisms, f^* is composition of the map f with the stalk map, while with pasting presheaves f_* is composition of the presheaf with f^{-1}. I guess a sheaf theorist might see both of those as images, and then “inverse” or “direct” just indicates variance with respect to the map f. My own focus on the local homeomorphisms, whatever its merits when investigating geometricity and arithmetic universes, leads me to disregard the pasting presheaves. All the best, Steve.
On 8 Nov 2022, at 10:20, Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt> wrote: Hi Steve,
The terminology is at odds with the localic case, but it seems to be standard in sheaf theory, independently of category theory and toposes: https://en.wikipedia.org/wiki/Direct_image_functor Then a question is what a left adjoint to an inverse image functor ought to be called in general…
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